PSI - Issue 24

Giovanni Zonfrillo et al. / Procedia Structural Integrity 24 (2019) 470–482 G. Zonfrillo and M.S. Gulino / Structural Integrity Procedia 00 (2019) 000–000

479

10

Table 3. Number of materials on which the remarkable models of Table 3 are based and related F 1 scores for each behaviour category. Features Softening Hardening Mixed Stable Total

Number of materials

72

59

5

7

143

f 1 = ln( f ) f 2 = ln( α ) f 3 = ln( σ y ) f 1 = ln(K) f 2 = ln(n) f 3 = ln( σ u )

F 1 score

93.1% 93.1%

84.7% 84.7%

0.0% 14.3% 82.5% 0.0% 14.3% 82.5%

Correct predictions Number of materials Correct predictions Number of materials Correct predictions Number of materials Correct predictions Number of materials F 1 score F 1 score F 1 score

98

99

33

8

238

91.8% 91.8%

82.8% 82.8%

43.1% 0.0% 78.2% 42.4% 0.0% 78.2%

72

59

5

7

143

f 1 = f f 2 = K f 3 = n

94.4% 94.4%

85.5% 84.7%

0.0% 0.0% 82.5% 0.0% 0.0% 82.5%

98

99

33

8

238

f 1 = σ u f 2 = σ u / σ y

90.3% 89.8%

86.9% 86.9%

0.0% 0.0% 73.1% 0.0% 0.0% 73.1%

99

102

33

8

242

f 1 = ln(K) f 2 = ln(n)

F 1 score

88.8% 87.9%

80.8% 80.4%

36.9% 0.0% 74.8% 36.4% 0.0% 74.8%

Correct predictions

where TP represents, for a specific category of material, the number of true positive predictions, FN the number of false negative predictions and FP the number of false positive predictions; these individual concepts are generally summarised in the so-called “confusion matrix”. In the specific problem, S represents the ratio between the frequency of threshold exceeding which leasd to correct classification of the material and the total number of materials with such behaviour within the sample; R instead represents the ratio between the frequency of threshold exceeding which leads to correct classification of the material and the number of times such behaviour is predicted. Finally, the so-called F 1 score (representing the harmonic mean of S and R ) allows assessing the actual predictive ability for the specific combination of model and threshold considered. Table 3 shows, for the models highlighted in Table 2, the number of materials on which these models are based and the associated F 1 scores for each material category. First, models have been excluded for which the combination of features is not available for at least 100 materials. The highest goodness-of-fit results from an imposed value of T equal to 0.4, for all derived logit models; as shown in Table 3, considering logarithmic values of the features often leads to better predictive abilities. Starting from F 1 scores, an inappropriateness of all the derived models in predicting both stable and mixed behaviour is also highlighted. Based on the data reported in Table 3, the model for which f , K and n are the considered features can be pointed out as the best one in terms of softening behaviour prediction (94% correctly predicted). Although this model can be extremely useful if applied during the design phase, it is clear that considering di ff erent features as ln(K) , ln(n) and ln( σ u ) results in a model with comparable goodness-of-fit, but more robust as a consequence of the higher number of materials used for the calculation (143 against 238). The predictive ability on the entire sample for this latter model is lower, also due to a lower predictive ability in terms of hardening behaviour. To highlight the predictive abilities of both models, Fig. 7 shows the distribution of the sample materials based on the values of f , K , n (Fig.7a) and ln(K) , ln(n) , ln( σ u ) (Fig. 7b), as a function of the materials’ behaviour; green-coloured points represent correct predictions of material behaviour, red-coloured points represent incorrect predictions. Considering the whole set of materials, the aforementioned combination of ln(K) , ln(n) , ln( σ u ) turns out to be the absolute best, if features are excluded whose data are not available for almost all materials collected in the database. Other combinations of three features in which at least one of the two “plastic” parameters of the Ramberg-Osgood re lation ( K , n ) is present perform similarly, as well as the combination of K and n which correctly predicts the behaviour of 75% of the sample. Nevertheless, the values of K and n are often absent in databases collecting material properties, available from web sources or technical literature. Excluding these parameters, the best performances are exhibited by combinations between σ u and σ y and in particular by those based on σ u and σ u / σ y . A prediction based on this logit 4. Discussion